Jean-Luc Brylinski

Lie theory and geometry : in honor of Bertram Kostant

This volume, dedicated to Kostant on his 65th birthday, is a collection of 22 papers by international leaders in the fields of Lie theory, geometry, algebra and math physics. The papers broadly reflect the range of Kostants work.

Geometric Construction of Quillen Line Bundles

The purpose of this paper is twofold. First we want to develop further and more systematically a cohomology theory adapted to holomorphic vector bundles with hermitian metrics, which in low degrees was introduced in joint work with McLaughlin [Br-ML2] [Br-ML3]Second we pursue cohomological and geometric definitions of the Quillen metric on determinant line bundles.

Advances in Geometry

This collection of invited mathematical papers by a list of distinguished mathematicians is an outgrowth of the scientific activities at the Center for Geometry and Mathematical Physics of Penn State University. The articles in this text present results or discuss perspectives on work that should be of interest to researchers and graduate students working in symplectic geometry and geometric quantization, deformation quantization, non-commutative geometry and index theory, quantum groups, holomorphic algebraic geometry and moduli spaces, quantum cohomology, algebraic groups and invariant theory, and characteristic classes.

Degree 3 Cohomology: Sheaves of Groupoids

We develop the theory of Dixmier-Douady sheaves of groupoids and relate it to degree 3 cohomology with integer coefficients. In §1 we explain the theory of descent for sheaves, based on local homeomorphisms. In §2, we introduce sheaves of groupoids (also called stacks) and gerbes. We relate gerbes on X with band a sheaf of abelian groups A with the cohomology group H 2(X, A). A gerbe with band ℂ X* is called a Dixmier-Douady sheaf of groupoids. In §3 we introduce the notion of connective structure and curving for such sheaves of groupoids; we obtain the 3-curvature Ω, which is a closed 3-form such that the cohomology class of \(\frac{\Omega }{{2\pi \sqrt { - 1} }}\) is integral. We prove that any 3-form with these properties is the 3-curvature of some sheaf of groupoids, and relate this to the constructions of Chapter 4. In §4 we use the path-fibration to define a canonical sheaf of groupoids over a compact Lie group. In §5 we give other examples of sheaves of groupoids connected with Lie group actions on a smooth manifold and with sheaves of twisted differential operators.

The Dirac Monopole

We begin by reviewing Dirac’s treatment of the magnetic monopole. While we will take a brief look at some of the basic ideas of quantum mechanics, very few details are actually needed for Dirac’s elegant derivation. Our treatment of the basics is close to [L-L], with all the physics carefully eliminated.

Complexes of Sheaves and their Hypercohomology

The word sheaf is a translation of the French word faisceau. Sheaves were first introduced in the 1940s by H. Cartan [Ca] in complex analysis (following works by K. Oka [Ok] and H. Cartan himself), and by J. Leray in topology [Le]. Since then sheaves have become important in many other branches of mathematics. Whenever one wants to investigate global properties of geometric objects like functions, bundles, and so on, sheaves come into play. There are many excellent introductions to the theory of sheaves [Go] [Iv] [B-T]; we will develop here only those aspects which will be useful throughout the book. We will begin with the notion of presheaf.

Line Bundles over Loop Spaces

The theme of this chapter is that a Dixmier-Douady sheaf of groupoids with connective structure over a manifold M leads naturally to a line bundle over the free loop space LM. This is in complete analogy to the well-known fact that a line bundle with connection over M leads to a function over LM, its holonomy.

Degree 3 Cohomology: The Dixmier-Douady Theory

Much of this book is devoted to a geometric description of the degree 3 cohomology H 3(M,ℝ) of a manifold M. Recall in the case of degree 2 cohomology, a theorem of Weil and Kostant (Corollary 2.1.4) which asserts that H 2(M,ℝ) is the group of isomorphism classes of line bundles over M. Moreover, given a line bundle L, the corresponding class c 1(L) in H 2 (M,ℝ) is represented by \(\frac{1}{{2\pi \sqrt { - 1} }}\) · K, where K is the curvature of a connection on L. We wish to find a similar description for H 3(M,ℝ), which is a more difficult task. One theory, due to Dixmier and Douady [D-D], involves so-called continuous fields of elementary C*-algebras. We will describe this theory in the present chapter and develop, in particular, the notion of curvature, which will be a closed 3-form associated to such a field of C*-algebras.